Objects in the middle of plane on hair particle system
The origin of the cubes in your group is in the center each cube instead of their respective bottoms. In each of the cubes in the group, move the origin to the bottom face to fix it:

Edit mode --> Select face --> Shift + S --> Snap cursor to selected

Then:

Object mode --> Ctrl + Alt + Shift + C --> Origin to 3D cursor

Another explanation on how to set the object's origin.